Classical Element Feedback Control for

Spacecraft Orbital Maneuvers

Bo J. Naasz∗ and Christopher D. Hall†

Aerospace and Ocean Engineering

Virginia Polytechnic Institute and State University

Blacksburg, Virginia 24061

Abstract

In this paper, we develop a nonlinear Lyapunov-based control law and a novel mean-motion

control strategy. This control is specifically intended for spacecraft orbital maneuvers and for-

mation flying applications. We apply the control to formation-establishment and formation-

keeping maneuvers for a proposed NASA mission. To provide target orbit states for feedback

control, we develop and apply an algorithm to calculate a formation master orbit represent-

ing the geometric center of the formation. We also define a new technique for choosing

nonlinear feedback control gains which appropriately scales the gains for orbital maneuvers.

The orbital element feedback control law, augmented by mean motion control, and applied

with appropriate gains, forces asymptotic convergence to a spacecraft target orbit, for a large

variety of spacecraft maneuvers.

Introduction

Spacecraft flying in close proximity is not a new concept. Gemini VI accomplished the first

space rendezvous on December 15, 1965, and performed station-keeping with Gemini VII over

three orbits at distances of 0.3 to 90 meters. Gemini VIII followed with the first docking of

two spacecraft in orbit on March 16, 1966. The first automated rendezvous was performed by

Soviet spacecraft Cosmos 186 and Cosmos 188 on October 27, 1967, when the two unmanned

∗Graduate Research Assistant. Currently with NASA Goddard Space Flight Center, Flight Dynamics

Analysis Branch, Code 572 Greenbelt, MD 20771, (301) 2863819, bo.naasz@gsfc.nasa.gov†Associate Professor. Associate Fellow AIAA. (540) 231-2314, cdhall@vt.edu

1

spacecraft performed a preprogrammed closure and docking maneuver.1 Since the 1960s,

numerous manned and unmanned spacecraft have performed operations in close proximity,

mostly for the purpose of docking for re-supply or crew transfer, and observation, as in the

case of Shuttle fly-arounds of the International Space Station. Autonomous formation flight

became a reality for spacecraft on May 17, 2001, when Earth Observing-1 (EO-1) performed

its first autonomous maneuver to maintain a one minute in-track separation with Landsat-

7.2 These close proximity spacecraft encounters have paved the way for spacecraft formation

flight for use in a number of varied scientific, military, and satellite service operations.

Motivation

Unlike previous close proximity operations, which were brief, high-thrust encounters, forma-

tion flying spacecraft must maintain relative separations for extended lengths of time (their

entire lifetimes), while expending minimal amounts of fuel. The need to conserve fuel in

formation flying spacecraft has motivated a great deal of research directed at improving our

understanding of the long term relative motion of spacecraft formations. Equipped with a su-

perior model of differential perturbation effects, we can design relative formation geometries

which minimize the need for formation-keeping maneuvers.3,4

Obviously, the most effective method of conserving fuel on formation flying spacecraft

is to design relative trajectories which do not degrade in the presence of differential per-

turbations. Unfortunately, it is not possible or feasible to design every formation with only

fuel conservation in mind. At some point, the requirements of formation flying missions

will conflict with the requirements of invariant formation design, making orbital maneuvers

inevitable.

Problem Definition

Traditionally, spacecraft orbits are controlled by engineers on the ground, who command

the spacecraft to perform thrust profiles designed to force optimal maneuver trajectories.

2

Linear feedback control is used primarily to counter unexpected perturbations, and main-

tain precalculated optimal trajectories. We develop nonlinear feedback control algorithms to

autonomously maneuver and maintain spacecraft formations, without the use of computa-

tionally expensive optimization techniques. These control laws should be applicable to real

satellites, and therefore must work in the presence of constraints such as thrust magnitude

and direction limitations. They should also seek to minimize fuel consumption, both of the

individual formation flyers, and of the formation as a whole. The formation flying strate-

gies should cover issues including the definition of reference, or target trajectories, and the

definition of appropriate nonlinear feedback control gains. Intuitively, the use of nonlinear

control in place of linear control should result in significant fuel savings. This savings should

be particularly evident when feedback control is used to perform large maneuvers, where the

nonlinear dynamics effects become pronounced.

This research is motivated by Leonardo, a proposed NASA mission where formation

flying is used to enhance the scientific mission of measuring the bi-directional reflectance

distribution function.5

One proposed Leonardo formation includes six spacecraft in low Earth orbits with incli-

nations varying from about 0.5 to 5.0 degrees, with right ascension and argument of latitude

variations resulting in satellite separation distances on the order of hundreds of kilometers.

Additional information on the Leonardo mission may be found in Ref. 6.

Formation Control Literature Review

Control of satellite formations can be considered in terms of two operational modes:

formation-keeping, and formation-maneuvering. The difference between the two modes is,

at first, subtle. It is not difficult to think of formation-keeping as a subcategory of formation-

maneuvering. However, when we consider the application of orbit control to establish large

formations, or to reconfigure them (formation-maneuvering), as compared to the mitiga-

tion of differential perturbations to maintain formation geometries (formation-keeping), the

distinction becomes more clear.

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Formation-keeping controllers have been developed by Vassar and Sherwood,7 for two

satellites in a circular orbit, Leonard, Hollister, and Bergmann,8 for two satellites with drag

panels to generate differential drag, and Middour,9 for along-track formation-keeping of low

eccentricity formation flyers.

A more recent formation-keeping control, the Folta-Quinn (FQ) Formation Flying Algo-

rithm10 uses Lambert’s two point boundary value problem and the ‘C*’ guidance and nav-

igation matrix to maintain a formation of two spacecraft. Carpenter, Folta and Quinn10,11

use the FQ algorithm, in conjunction with a linear-quadratic-Gaussian (LQG) control12 to

develop a decentralized control for autonomous formation flying for use in the New Mille-

nium Program and the Earth Observing-1 (EO-1) mission. The formation model includes

two spacecraft, one with real-time GPS and autonomous control (EO-1), and one with orbit

determination and maneuver generation performed on the ground (Landsat-7). The au-

thors conclude that autonomous, decentralized control is not only feasible, but beneficial

over traditional orbit control procedures. The preliminary results2 of the EO-1 experiment

show that the FQ formation flying algorithm has succeeded in providing robust, closed-loop,

autonomous control of NASA’s first formation flying spacecraft.

Formation-maneuvering is especially important in establishing a formation, but may

also be used to modify a formation based on changing mission requirements. Schaub and

Alfriend13,14 use mean orbital element differences to control the relative orbits of formation

flying spacecraft with identical ballistic coefficients. The authors’ use of mean orbital element

differences stresses long-term behavior over short-term deviation, by forcing the control to

compensate for secular and long period drifts, while ignoring short period oscillations. A

modified form of Gauss’ variational equations of motion is used to develop a sequential,

impulsive algorithm to correct orbit element errors. The control law establishes the desired

formation in a nearly fuel-optimal sense by requesting impulses in the polar and equatorial

regions of the orbit, only. This impulse timing allows correction of each element, with

minimal impact on the other elements. The authors also develop a cartesian, Lyapunov-

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control law, and apply it using relative position and velocity tracking vectors calculated

from mean orbital elements. A comparison of the two nonlinear feedback control laws shows

that the cartesian form is effective for short time-frame, high thrust maneuvers, while the

orbital element form performs best in multiple-orbit or low-thrust maneuvers.

Ilgen15 develops Lyapunov-optimal feedback control laws using Gauss’s form of La-

grange’s planetary equations (LPE) in classical and equinoctial orbital-element forms. The

author presents equations of motion for the equinoctial elements in terms of thrust compo-

nents in the equinoctial coordinate axes. While the author does not discuss formation-flying

control directly, the paper provides an excellent starting point for the development of an

elemental, Lyapunov-optimal, formation flying feedback control law.

In Ref. 16 the authors develop non-impulsive control for transfers between elliptic orbits,

using feedback of the angular momentum vector, L, and the Laplace (or eccentricity) vector,

A. The authors suggest the algebraic structure of these vectors leads to superior control as

compared with elemental feedback control. The authors do not discuss gain selection, nor do

they comment on controlling the angular position within an orbit. Future work suggestions

include investigation of J2 effects, and optimization through gain selection. While the authors

do not address formation-flying control directly, it may be possible to expand their approach

to include an in-orbit angular position parameter, and thus control the full six state feedback

control problem.

A more thorough review of formation flying literature may be found in Ref. 17.

Approach

We approach formation flying control in two steps: target orbit definition; and target orbit

rendezvous. We assume desired relative formation geometries are known, and most likely

chosen to optimize the performance of mission operations other than formation control. In

the case of the Leonardo mission, the relative formation geometries have been designed by

NASA scientists to optimize the mission performance.? If possible, these relative geometries

should also be designed to minimize the relative drift of the formation.

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To maintain and maneuver a formation of spacecraft, we first develop a method for

defining the absolute state of the formation, and the relative states of the spacecraft within

the formation. We define the absolute state of the formation in terms of the osculating

elements of some “master” orbit, positioned in the geometric center of the desired formation.

From this master orbit, we define differential orbital element sets, used to calculate the

desired spacecraft states at any given time. During simulation, we integrate the individual

spacecraft states, as well as the master orbit state, and calculate the target orbits at each

time step by adding constant differential orbital element sets to the formation master orbital

elements.

To force convergence of the formation flying spacecraft to the desired orbit states, we

develop a nonlinear orbit control law which uses classical orbital element error feedback,

and mean motion control to force global, asymptotic convergence to the desired spacecraft

states. We present the results of control law application in formation-establishment and

formation-keeping maneuvers for the Leonardo formation.

Model and Equations of Motion

The dynamics of space objects is most often described by two-body motion in a Keplerian

gravitational field, with corrections to compensate for forces from perturbations such as non-

spheroidal central body effects, atmospheric drag, solar and lunar gravitational effects, and

solar radiation pressure. In this section, we present a brief description of Keplerian orbits,

and the effect of some perturbations on formation dynamics. We conclude the section with

a discussion of some formation flying concepts, including formation models and formation

master orbit definitions.

In cartesian form, the equations of motion for a point-mass satellite take the form

~r = − µ

‖~r‖3~r+ ~ap (1)

where ~r is the position vector of the satellite measured from the center of mass of the

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primary body, µ is the gravitational parameter of that body, and ~ap includes perturbation

accelerations caused by non-spheroidal gravitational effects of the central body, atmospheric

drag, third body effects and so on. If ~ap = ~0 then the equations of motion reduce to the ideal

Keplerian situation where the central body is a perfect sphere and all other disturbances are

zero.

We can also express the motion of a point-mass satellite in terms of orbital elements.

We define the classical orbital elements,18 as the semi-major axis, a, the eccentricity, e, the

inclination, i, the right ascension of the ascending node, Ω, the argument of periapse, ω,

and the mean anomaly, M . We also need to refer to the true anomaly, ν, and the argument

of latitude, θ = ω + ν. The equations of motion of a controlled spacecraft in terms of the

classical orbital element set are given by Gauss’s form of Lagrange’s planetary equations:19

da

dt=

2a2

h

(

e sin ν ur +p

ruθ

)

(2)

de

dt=

1

hp sin ν ur + [(p+ r) cos ν + re] uθ (3)

di

dt=

r cos θ

huh (4)

dΩ

dt=

r sin θ

h sin iuh (5)

dω

dt=

1

he[−p cos ν ur + (p+ r) sin ν uθ]−

r sin θ cos i

h sin iuh (6)

dM

dt= n+

b

ahe[(p cos ν − 2re)ur − (p+ r) sin ν uθ] (7)

where ur, uθ, and uh are the radial, transverse, and orbit normal control, or disturbance

acceleration components, n is the mean motion, and p, h, and b are the semi-latus rectum,

the angular momentum, and the semi-minor axis, respectively.

Perturbation Effects

Whereas orbital perturbations have a significant impact on the absolute motion of individual

spacecraft, the effect of perturbations on the relative motion of formation flying spacecraft

can be quite limited. For formations in Earth orbit, the two most significant perturbations

7

are Earth oblateness effects, and atmospheric drag effects.

For formations of identical spacecraft (equal size and mass), the differential drag effect is

negligible, and the most significant perturbation to the formation geometry is the differential

J2 effect. Earth oblateness causes regression of the node, or rotation of the orbit about the

inertial i3 vector, as well as secular rotation of the orbit about the orbit normal vector. The

secular drifts of Ω, ω, and M due to J2 perturbations are:

dΩ

dt= −3nJ2

2

(

r⊕p

)2

cos i (8)

dω

dt=

3nJ24

(

r⊕p

)2

(5 cos2 i− 1) (9)

dM

dt=

3nJ24

(

r⊕p

)2√1− e2 (3 cos2 i− 1) (10)

where r⊕ is the radius of the Earth. Notice that the secular drift due to J2 effects varies

with inclination. Formations of spacecraft with no inclination variation will drift as a group,

experiencing no relative deformation. Formations with inclination variations encounter dif-

ferential secular drift, which must be mitigated to maintain the formation geometry. A more

detailed description of formation perturbation effects is available in Ref. 20.

Formation Flying Concepts

In this paper, we study the control of a formation of six identical spacecraft. We calculate the

desired orbits of the maneuvering spacecraft from a central reference orbit, which we refer to

as the master orbit. We refer to the desired orbits as target, or virtual orbits. We treat each

maneuver as a rendezvous between a maneuvering spacecraft, and a passive, target orbit, or

virtual spacecraft. We now introduce some basic formation geometries and models, and a

method for calculating the formation master orbit and desired spacecraft orbits.

Formation Models

Figure 1 illustrates three models for controlling and simulating formation dynamics. The

three models increase in complexity from a simple target-interceptor problem to cooperative,

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multi-craft maneuvering problem. The target-interceptor problem simply involves maneu-

vering one spacecraft to rendezvous with another.

The intermediate model is a leader-follower (or master-slave) type formation where the

desired orbit of the active spacecraft (the follower) is determined by a formation planner

from the leader’s orbit. In the case of an anomaly shift formation, the active spacecraft tar-

gets a virtual orbit defined by a δν shift of the leader’s orbit. This model is only required if

differential perturbations, such as Earth oblateness or orbit control at the leader spacecraft,

are applied, causing the virtual target orbit to drift with respect to leader’s orbit. If no per-

turbations are applied in orbit propagation, this model is identical to the target-interceptor

problem, as the virtual target orbit follows a path identical to the leader’s.

The cooperative, multi-craft maneuvering problem involves the complicated task of deter-

mining target orbits for multiple, active spacecraft from the inertial orbits of the spacecraft,

and a predetermined relative formation type. In Ref. 21, the authors show that the use of

two actively controlled spacecraft as opposed to a leader-follower type formation architec-

ture improves the efficiency of formation control by minimizing both the time, and the total

number of thrust impulses required to establish the desired formation.

The desired relative orbit can be chosen in many ways, possibly to optimize the mission

performance in some sense,22 or to minimize the effect of gravitational perturbations on the

formation geometry3 in an effort to minimize formation-keeping propellant consumption.

For this study, we assume that the formation geometry is already defined, and seek to find

ways to model, maneuver, and maintain that formation geometry.

The Master Orbit

To implement the cooperative formation control problem we must develop a method for

calculating the desired orbits of all of the spacecraft in the formation as a function of the

current location of each spacecraft, and the required formation geometry. We would like to

define a master orbit, close to the center of the cluster, for use as a reference point in the

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calculation of the individual satellite target orbits. Starting with the position and velocity

vectors of each of the formation flyers, and the required relative geometry, we can calculate

the individual reference orbits in three steps: 1) initialize the formation; 2) calculate the orbit

which remains closest to the center of the formation throughout the formation’s motion (the

master orbit); 3) define orbital element offsets used to calculate the desired orbits from the

master orbit.

Initializing the Formation. Formation geometries are often defined in relative terms,

such as differential orbital elements, and leave the absolute positioning general. We study

the control of a sample formation for the Leonardo mission? which can be fully described in

terms of some absolute orbital elements, and some relative, or differential, orbital elements.

The sample Leonardo formation, defined by Table 1, consists of six spacecraft in 400 km,

circular orbits, with varying inclinations, node vectors, and arguments of latitude. Note that

in this formation, the inclination varies between spacecraft, but is still given in absolute

terms.

This formation is defined in terms of both relative and absolute parameters. The values

of Ωj and θj are not important, as long as their relative values match those given in the

table. To initialize the formation, we choose reference values of Ω0 and θ0 at random, and

construct the individual orbits from the defined offset values.

Defining the Master Orbit. Once the formation has been initialized, it is convenient

to define a master orbit which remains in the center of the formation as the spacecraft

complete their orbits. For simple formations such as the anomaly shift formation, where the

argument of latitude, θ, is the only parameter that differs between spacecraft, this master

orbit is easily found. Since five of the orbital parameters are shared by all spacecraft in the

formation, the sixth, the argument of latitude of the master, θm, can be found by averaging:

θm =1

N

N∑

j=1

θj (11)

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For more complex formations such as the one described in Table 1, the definition of a

central master orbit requires a more robust method of calculation. We derive this method

by considering some conserved properties of the formation: the mean angular momentum

vector, and the formation equivalents of the node and eccentricity vectors.

For a single spacecraft, the orbital elements are calculated from the position and velocity

vectors by manipulation of these three vectors. The orbital elements of the master orbit can

be found from the position and velocity vectors, rj and vj, of individual formation flyers in a

similar way, using the three fundamental vectors of the cluster: the mean angular momentum

vector, h, the mean node vector, n, and the mean eccentricity vector, e.

Algorithm OrbitMaster

Following the same basic steps used to calculate the orbital elements of a single satellite from

position and velocity vectors, we calculate the master orbit elements for a formation of N

spacecraft as follows:

1. Calculate the mean angular momentum vector, h,

h =1

N

N∑

j=1

rj × vj =1

N

N∑

j=1

hj (12)

2. Calculate the mean node vector, n,

n =1

N

N∑

j=1

K× hj =1

N

N∑

j=1

nj (13)

3. Calculate the mean eccentricity vector, e,

e =1

N

N∑

j=1

1

µ

[(

v2j −µ

rj

)

rj − (rj · vj)vj]

=1

N

N∑

j=1

ej (14)

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4. Calculate the inclination of the master orbit, im,

cos im =h · i3h

im ∈ [0, 180] (15)

5. Calculate the right ascension of the ascending node of the master orbit, Ωm,

cosΩm =n · i1n

if n · i2 > 0,Ωm < 180 (16)

6. Calculate the argument of periapsis of the master, ωm (assuming e is non-zero),

cosωm =n · ene

if e · i3 > 0, ωm < 180 (17)

7. Calculate the true anomaly of the master, νm (assuming e is non-zero),

cos νm =e · rer

if r · v > 0, ν < 180 (18)

where r is the component of the mean radius vector, rmean, perpendicular to the mean

angular momentum vector,h:

rmean =1

N

N∑

j=1

rj (19)

r = rmean − (rmean · h)h (20)

8. Calculate the argument of latitude of the master, θm, the angle between n and r,

cos θm =n · rnr

if r · i3 > 0, θm < 180 (21)

Calculating the Desired Orbits. Given these definitions for the master orbit elements,

we calculate the desired states of the formation flyers at any time by propagating the master

orbit to the desired time, and adding orbital element offsets slightly modified from those

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in Table 1. These new offsets, ∆Ωj and ∆θj, are calculated by subtracting the initialized

spacecraft orbital element values from the master orbital elements:

∆Ωj = Ωj − Ωm, ∆θj = θj − θm (22)

We calculate the desired orbit of the formation flyers from these new offsets as in Table 1,

except that we use the master orbit for the reference values of Ω and θ.

Example: Master Orbit Calculation. To calculate the master orbit of a cluster of

spacecraft in the Leonardo formation, we choose reference values of Ω0 = 0 and θ0 = 0. For

these reference values, the orbital elements of the Leonardo spacecraft are given in Table 2,

along with the corresponding master orbit elements.

As shown in the table, the orbit that best defines the center of the formation has an

inclination of about 0.25, and right ascension and argument of latitude of about 98 and

266, respectively. Notice that these values are far from the average values, and that the

master inclination is nearly zero. This result is not surprising, as the orbital planes of the

individual spacecraft are fairly evenly dispersed around the equator by varying values of Ω.

With the master orbit identified, we can calculate the orbital element offsets in Eq. (22) by

subtracting the initialized spacecraft orbital element values from the master orbital elements.

The new offsets can then be used to calculate the desired state of any of the spacecraft in the

formation at any time by propagating the master orbit, and shifting the Ωj and θj values.

Note that these new offsets are not the same as those given in Table 1, and apply only to

the formation generated using the reference values Ω0 = 0 and θ0 = 0.

The definition of a central formation master orbit is essential to formation flying opera-

tions. We calculate the target spacecraft orbits as a function of the formation master orbital

elements, and predefined formation geometries. We also use the master orbit calculation

algorithm to re-center the formation after periods of differential drift due to differential per-

turbation effects. The master orbit defined in this section is one of many possible choices for

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the formation master. In a subsequent section, we show how other formation master orbit

definitions can result in improved fuel efficiency for the formation.

Lyapunov Control Law

Whereas linear control techniques can be highly effective in the control of nonlinear systems,

nonlinear control strategies can be beneficial in a variety of ways. For example, nonlinear

controllers may: a) increase the region of state space in which we can effectively control the

system, b) improve control robustness to parametric uncertainty, c) obtain truer optimality

results, d) enable control of systems which are not linearly controllable, and e) allow us to

preserve and exploit physical insight.

In this section we develop nonlinear, Lyapunov-based classical element feedback control

for spacecraft orbital maneuvers. The first part of this control law, based on previous work

by Ilgen,15 and comparable to work by Schaub and Alfriend,23 establishes the desired orbital

plane. The second part, which we call “mean motion control,” takes advantage of the natural

relative dynamics to control spacecraft position within the orbit.

Classical Orbital Element Control

The equations of motion (2 – 7) can be written as:

oe = f(oe) + G(oe)u (23)

where oe is the vector of orbital elements, [ a e iΩωM ]T , and u is the vector of controls,

[ur uθ uh ]T .

We separate these equations of motion into two systems: one system consisting of the

first five elements, which define the shape, size, and orientation of the orbit, and the second

system consisting of the in-plane angular position of the spacecraft.

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For the first system, we define the error η as

η =

a− a∗

e− e∗

i− i∗

Ω− Ω∗

ω − ω∗

=

δa

δe

δi

δΩ

δω

(24)

where (·)∗ is the target element.

The equations of motion for this system are:

η = Gu (25)

where the input matrix, G, from Eqs. (2–7), is

G =

2a2esνh

2a2phr

0

psνh

(p+r)cν+reh

0

0 0 rc(ω+ν)h

0 0 rs(ω+ν)hsi

−pcνhe

(p+r)sνhe

− rs(ω+ν)cihsi

ur

uθ

uh

(26)

where the sine and cosine functions are abbreviated as s(·) and c(·).

A candidate Lyapunov function for this system is

V (η) =1

2ηTη (27)

whose time derivative is given by

V (η) = ηT η = ηTG(η) u (28)

15

Choosing the control, u, as

u = −GTKη (29)

where K is a positive definite gain matrix, results in

V = −ηTGGTKη ≤ 0 (30)

The time derivative of the Lyapunov function is negative semi-definite. To prove asymp-

totic stability, we apply LaSalle’s invariance theorem. The time derivative of the Lyapunov

function is always zero when η = 0, and could be zero when the trigonometric functions of

θ = ω + ν are zero, which occurs when θ = kπ/2, where k is an integer. The set where

θ = kπ/2 is not an invariant set, because θ is time-varying and therefore trajectories that

start in the set do not remain in the set. Therefore, the largest invariant set where V = 0 is

the set η = 0. So by LaSalle’s invariance principle, the system is asymptotically stable under

the choice of control given in Eq. (29). Furthermore, the system is globally asymptotically

stable because V →∞ as |η| → ∞.

Choosing a diagonal, positive definite gain matrix, K, the control law from Eq. (29) is:

ucl = −G(η)TKη = −

2a2esνh

2a2phr

0

psνh

(p+r)cν+reh

0

0 0 rc(ω+ν)h

0 0 rs(ω+ν)hsi

−pcνhe

(p+r)sνhe

− rs(ω+ν)cihsi

T

Kaδa

Keδe

Kiδi

KΩδΩ

Kωδω

(31)

The angle errors δΩ, δω, and δM are measured from parameters defined between 0 and

2π. To ensure proper error feedback, these angular errors should be “short-way” angle

measurements, defined between −π and π.

Mean Motion Control

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To control the in-plane angular motion of a spacecraft, we take advantage of a useful natural

component of the nonlinear dynamics: the differential mean motion, δn. Observe that an

effective method for correcting in-track errors is to force a semi-major axis error, changing

the mean motion such that the in-plane angle is corrected. For an uncontrolled spacecraft

in two-body motion, the orbital element dynamics simplify to

dM

dt= n (32)

and thus the relative dynamics can be written as

d

dt(δM) =

dM

dt− dM∗

dt= n− n∗ = δn (33)

where

δn =

√

µ

a3−√

µ

a∗3(34)

To drive the mean anomaly error, δM , to zero, we use

˙δM = −KnδM (35)

whereKn is a positive gain. Combining Eqs. (34) and (35), and using canonical units (µ = 1),

we obtain

−KnδM =

√

1

a3−√

1

a∗3(36)

Solving for a, we define a new target semi-major axis, a∗∗, which forces the mean anomaly

error to zero:

a∗∗ =(

−KnδM +1

a∗3/2

)−2/3

(37)

Notice that as δM goes to zero, the mean motion control target semi-major axis approaches

the original target value, a∗. In application, we replace the mean anomaly error, δM , in

Eq. (37) with argument of latitude error, δθ, so that the mean motion control properly

17

positions the spacecraft within the orbital plane, even in the presence of argument of periapse

error.

Using the orbit control law defined in Eq. (31), with the target semi-major axis a∗∗ defined

in Eq. (37), we can control the full, nonlinear motion of an orbiting spacecraft.

Control Application Issues

To apply the control laws developed in this paper, we must first address some control appli-

cation issues. Real spacecraft have numerous constraints limiting the application of thrust.

It is not always possible for thrusters to deliver the exact magnitude and direction of thrust

requested by the control law. In this section, we discuss the impact of constrained thrust

magnitude and direction on control law design, as well as a method for choosing appropriate

gains for the nonlinear control.

Constrained Thrust Magnitude. Since most spacecraft have fixed magnitude thrusters,

we must develop some way of discretizing the control acceleration requested by the control

laws. For example, if the orbital element feedback control law requests acceleration of mag-

nitude A in the α direction, we must determine whether or not to thrust, and if so, in what

direction.

This decision is straightforward for this elemental control law, as we have full scaling

freedom in the choice of gains. We simply calculate a desired thrust direction and magnitude

from the control law, and thrust if and only if the desired thrust magnitude is greater than

the available thrust magnitude. Mathematically, the thruster on/off logic requests thrust in

the α direction if and only if A > T/m where T is the available thrust magnitude, and m is

the spacecraft mass.

Constrained Thrust Direction. In the case of the HokieSat mission, we must not only

scale the thrust magnitude, but also constrain the thrust direction.? HokieSat’s orbit control

consists of four pulsed plasma thrusters (PPTs?) aligned in the local horizontal frame, with

no thrust available in the radial direction.

18

The classical feedback control laws can be modified to exclude radial thrusting by setting

the terms in the first column of theG matrix in the control definition to zero, thus requesting

only transverse and orbit normal thrust.

Circular Orbits. While the equinoctial control law presented in Ref. 17 avoids the

singularities associated with circular and equatorial orbits, the lack of physical meaning

makes the equinoctial elements difficult to visualize. In the case of circular orbits, the

argument of periapse and the mean anomaly are undefined, and singularities appear in the

classical element equations of motion of those variables. We can modify the classical orbital

element Lyapunov control to deal with these issues by: 1) setting the gain Kω to zero so

that the errors associated with that gain do not contribute to the total requested thrust; 2)

using the argument of latitude error, δθ, in place of δM in the mean motion control. These

modifications are acceptable as long as the eccentricity error remains small, so that δθ is a

good approximation of the in-plane-angle error.

Gain Selection

Intuitively, nonlinear control of formation flying spacecraft should provide improvements

over more traditional, linear feedback control techniques such as LQR and H∞. One major

obstacle in the use of nonlinear controls such as the Lyapunov-based control developed in

this paper is the lack of a well-defined method for choosing gains. Whereas numerous tools

exist for gain selection for linear feedback control methodologies, nonlinear gain selection

techniques are often limited to trial-and-error type approaches. In the full state feedback

problem, the orbital element feedback control laws require a [6× 6] gain matrix. Even using

only the diagonal terms, we are unlikely to find six effective gains using a trial-and-error

approach.

To motivate a more effective gain selection process for the full-state feedback problem, we

explore the use of the nonlinear control laws in more limited control problems. The orbital

element Lyapunov control law can be used to control specific elements, by setting the gains

corresponding to the uncontrolled elements to zero. For example, the Lyapunov orbit control

19

law can be used to perform a circle-to-circle orbit-raising maneuver using feedback of the

semi-major axis error, δa, and the eccentricity error, δe, as in

ucl = −

2a2esνh

2a2phr

psνh

(p+r)cν+reh

T

Kaδa

Keδe

(38)

The selection of gains for this problem is simple, as there are only two gains to choose.

For a sample orbit-raising maneuver from 7,000 km to 10,000 km, using 1 N thrusters fired

continuously, trial-and-error tuning yields gains Ka = 0.7 and Ke = 1.174.

We can expand this problem to include inclination feedback by using

ucl = −

2a2esνh

2a2phr

0

psνh

(p+r)cν+reh

0

0 0 rc(ω+ν)h

T

Kaδa

Keδe

Kiδi

(39)

Figure 2 shows an example of this type of maneuver, in which the same circle-to-circle

orbit-raising maneuver is performed, but with an inclination shift from 28 to 60. Plot A of

the figure shows the maneuver performed with gains Ka = 1, Ke = 1, and Ki = 50. Plot B

shows the maneuver performed with gains Ka = 1, Ke = 1, and Ki = 5. Varying the gains

has a predictable effect on the maneuver: because the inclination is weighted more heavily in

plot A, the trend in the maneuver is to correct the out-of-plane error, and then complete the

orbit-raising. In plot B, where the semi-major axis and inclination are more equally weighted,

the control forces orbit-raising to occur first, and then completes the inclination maneuver.

The maneuver illustrated in plot A requires about 13% more fuel than maneuver B. This

fuel savings is easily explained by the fact that the ∆V required for simple plane change

maneuvers is directly proportional to the initial speed. Maneuver B saves fuel by performing

the majority of the plane change maneuver after it has already increased the semi-major axis

(and thus decreased the speed).

To choose appropriate gains for the full orbital element feedback problem, we strive

20

to understand the response of the individual elements to thrust impulses. The effect of

radial, transverse, and orbit-normal thrust impulses on the osculating orbital elements varies

throughout the orbit as a function of the true anomaly, as given by Gauss’s form of Lagrange’s

planetary equations. For example, the change in semi-major axis resulting from an impulsive

∆V = (T/m)∆t, in the radial and transverse directions, respectively is

∆ar =2a2e sin ν

h

T

m∆t (40)

∆aθ =2a2(1 + e cos ν)

h

T

m∆t (41)

where T , m, and ∆t are the thrust force, spacecraft mass, and thrust duration. The max-

imum change in the semi-major axis results from an impulse in the transverse direction at

periapse (where ν is 0):

∆amax =2a2(1 + e)

h

T

m∆t (42)

We use this knowledge as a guide for selecting the gain, Ka, associated with the semi-major

axis error feedback. From Eq. (31), the transverse Lyapunov control acceleration resulting

from a semi-major axis error, δa, is

uaθ(δa) = −(

2a2(1 + e cos ν)

h

)

Kaδa (43)

We choose Ka such that the maximum value of uaθ is equal to the available thrust

acceleration magnitude, in which case the control on-off logic asks for transverse thrust only

when the semi-major axis error δa is at least ∆amax. Furthermore, the control should thrust

in the presence of this error only once in the orbit, at the point where the control required

to correct the error is minimized. Mathematically, we choose the minimum value of Ka

such that the magnitude of the control given in Eq. (43) is equal to the available control

21

acceleration, T/m:

|uaθ(∆amax)| =∣

∣

∣

∣

∣

−(

2a2(1 + e cos ν)

h

)

Ka2a2(1 + e)

h

T

m∆t

∣

∣

∣

∣

∣

=T

m(44)

This minimum value of Ka occurs at periapse, and is given by

Ka =h2

4a4(1 + e)21

∆t(45)

With this choice, the thruster on-off logic requests thrust when the semi-major axis error is

greater than ∆amax, and does nothing when the error is less than ∆amax. Choosing all of

the gains in this way properly weights the elemental errors, and also prevents chatter in the

system, as the control law does not attempt to control elemental errors more precisely than

is possible with the available impulse.

With the exception of Kω and KM , the remaining control gains are easy to calculate

using the same method used to find Ka, and are given by:

Ke =h2

4p21

∆t(46)

Ki =

[

h+ eh cos (ω + arcsin(e sinω))

p(−1 + e2 sin2 ω)

]21

∆t(47)

KΩ =

[

h sin i(−1 + e sin (ω + arcsin(e cosω)))

p(1− e2 cos2 ω)

]21

∆t(48)

The maximum value of ∆ω occurs when transverse thrust is applied near ν = π/2.

Therefore, the appropriate selection of Kω comes from the transverse ω control equation:

uωθ(δω) = −p sin νhe

(

1 +1

1 + e cos ν

)

Kωδω (49)

Kω(ν) = min

e2h2 csc2 ν

p2(

1 + 11+e cos ν

)2

1

∆t

(50)

22

Equation (50) is minimized on ν when the function

f(ν) =csc2 ν

(

1 + 11+e cos ν

)2 (51)

is minimized, or the first derivative of f(ν) with respect to ν is zero, and its second derivative

is positive. The first derivative is

df

dν= −2 (1 + e cos ν) csc ν (e+ e (3 + e cos ν) cot2 ν + 2 cot ν csc ν)

(2 + e cos ν)3= 0 (52)

To find the roots of the first derivative, we expand g(ν), the numerator of f(ν), in a first-order

Taylor series about e = 0, and solve for the approximate root:

g(ν, e) = 2 (1 + e cos ν) csc ν(

e+ e (3 + e cos ν) cot2 ν + 2 cot ν csc ν)

(53)

g(ν, 0) = 4 cot ν csc2 ν (54)

Equation (54) is zero when ν = π/2. The first order approximation of the root is thus

ν ' π/2 + ν1e (55)

where ν1 is the root of the following equation

dg(π/2 + ν1e, e)

de

∣

∣

∣

∣

∣

e=0

= 4ν1 − 2 = 0 (56)

The approximate root of df/dν is

ν ' π

2+e

2(57)

23

Substituting into Eq. (50), we obtain the approximate minimum value of Kω:

min(Kω) 'e2h2 sec2 e

2

p2(

1 + 11−e sin e

2

)2

1

∆t(58)

Since the root is only valid to O(e), we expand Kω in a Taylor series expansion about e = 0,

and arrive at the final selection of the gain Kω:

Kω =e2h2

4p2

(

1− e2

4

)

1

∆t(59)

The last of the six gains, KM , found using the same Taylor series approximations, also

occurs at ν ' π2+ e

2, and is given by

KM =a2e2h2

4b2p2

(

1− e2

4

)

∆t (60)

We have defined a positive definite, diagonal gain matrix for use in classical orbital

element Lyapunov control that: 1) guarantees asymptotic convergence to an error envelope,

2) appropriately weights the elemental errors, and 3) eliminates the problem of chattering

by properly defining the size of the error envelope. The gains are defined at Eqs. (45), (46),

(47), (48), (59), and (60). These gains should be calculated as a function of the target orbit

at the beginning of a maneuver, and held constant throughout the maneuver.

Examples

In this section we

Sample Orbit Maneuvers

In addition to controlling the full orbital motion of a spacecraft, the orbital element feedback

control law can control individual elements of the orbit, as shown in subsequent examples

below. In any maneuver performed using the orbital element feedback control law, the

24

control gains should be scheduled so that the spacecraft first acquires the desired orbit,

and then establishes its angular position within that orbit. Gain scheduling is especially

important for spacecraft in nearly circular or nearly equatorial orbits, where small changes

in some orbital parameters (eccentricity and inclination) can result in large changes in other

elements (argument of periapse and right ascension). For spacecraft in more moderate orbits,

we can relax the gain scheduling requirement, but should never forget that often the angular

errors in the feedback loop are calculated from varying reference parameters.

Leonardo Maneuvers

Two maneuvers of interest to the Leonardo formation are the formation-establishment, and

formation-keeping maneuvers. In the formation-establishment maneuver, the spacecraft are

initially positioned in a parking orbit, and must use orbit control laws to initialize the

formation. In the formation-keeping maneuver, the spacecraft are initially in a perturbed

arrangement, and must re-establish the formation geometry using orbit control.

Leonardo Formation Establishment. The Leonardo formation-establishment prob-

lem involves the maneuvering of six spacecraft from a post-launch parking orbit into the

desired formation geometry. We assume the spacecraft are launched as a stack, and there-

fore share the same initial conditions. One choice for the initial parking orbit is the formation

master orbit defined in Table 2. This master orbit is the geometric center of the formation

and therefore a logical choice for the initial parking orbit.

Figure 3 shows a three-dimensional plot of the application of scaled classical orbital

element Lyapunov control to the formation-establishment problem, for one of the spacecraft

in the Leonardo formation. In the simulation, a 150-kg spacecraft is maneuvered from the

formation master orbit to the desired orbit defined in Table 2 as spacecraft 1, with thrust

magnitude constrained to 1 Newton. For this simulation the target orbit is nearly circular,

so the gain Kω is set to zero so that the argument of periapse error, δω (which is undefined

for circular orbits), does not contribute to the requested thrust. For the same reason, the

argument of latitude error, δθ, is used in the mean motion control.

25

For this simulation we use a mean motion control gain of Kn = 1 × 10−4 for the first

3.1 orbits, and Kn = 0.1 for the remainder of the simulation. Notice that the out-of-plane

error is corrected after just over three orbits, but the radial and in-track errors remain until

after the Kn gain is increased. This result is not surprising. The starting choice of Kn

creates a small semi-major axis error (negative δr3 represents error in the positive radial

direction). Close inspection of the δr1 data reveals that this small semi-major axis error

is gradually correcting the in-track error. If the simulation is allowed to run long enough,

without increasing Kn, the in-track error will eventually decrease to the error envelope, and

the semi-major axis error, shown by radial error, will be corrected. This approach provides

significant fuel savings, as the control law inherently allows drifting arcs in the maneuvering

spacecraft’s trajectory.

Figure 4 shows the values of the semi-major axis, a, the target orbit semi-major axis,

a∗, and the semi-major axis requested by the mean motion control, a∗∗, for the duration of

the maneuver. Recall that the elemental control law feeds back δa = a− a∗∗. In the initial

stage of the maneuver, the semi-major axis remains within a few kilometers of the target

value. Once the phasing part of the maneuver begins, the mean motion control forces δa

up to almost 20 kilometers. As the in-track error is corrected, the value of a∗∗ gradually

decreases to the original target value of a∗. Notice that after about 5.5 orbits, the value of a

starts to make steps towards the target value. These steps correspond to impulses correcting

the semi-major axis. The level areas between the step changes correspond to coasting arcs,

where no thrust is applied, as the in-track error is slowly corrected by a constant level of

semi-major axis error.

The maneuver shown in Figs. 3 and 4 requires a ∆V of approximately 375 m/s. To

establish the entire formation, using 1-N, scaled, classical element control, the individual

spacecraft require between 281 and 925 m/s, each. This large variation in fuel usage is an

important observation, as it means that during this maneuver the ballistic coefficients of

the spacecraft change non-uniformly throughout the formation. This non-uniform ballistic

26

coefficient distribution results in differential atmospheric drag on the formation, and should

be avoided.

While the formation-establishment maneuver is an interesting problem, low-thrust feed-

back control might not be the best choice for this stage of the mission. For a formation of

this size, a more practical approach might be to use a launch vehicle upper stage to place

each spacecraft in the stack, one at a time, on or near its desired orbit, as is commonly done

with spacecraft constellations. Of course, the use of an upper stage loses its appeal as the

difference between the orbital planes of the formation flyers increases. Ultimately, we want

to choose a method that minimizes the total propellant mass at launch, but also maintains

mass symmetry amongst the formation flyers, reducing the effect of differential drag.

Leonardo formation-keeping. The formation-keeping problem involves the mainte-

nance of the formation geometry in the presence of differential orbital perturbations such

as non-spherical central body effects, atmospheric drag, and solar radiation pressure. We

assume that the Leonardo spacecraft are, for the most part, identical in size and mass, and

therefore that the differential effects of atmospheric drag and solar radiation pressure are

negligible. This assumption will be invalid once the effects of formation geometry on the pro-

pellant mass required by each spacecraft are considered; however, the assumption is useful

in initial studies of formation-keeping requirements.

As shown in Eqs. (8–10), the magnitude of J2 secular drift varies with inclination. Because

the Leonardo spacecraft have different inclinations, the formation will experience differential

secular drift due to Earth oblateness effects.

While it may be ideal, from a control engineer’s perspective, to correct orbital errors

resultant from perturbation effects as they appear (indeed, a simple way to do this is to

simply allow the spacecraft to thrust whenever the thruster on/off logic requests thrust),

other spacecraft operations, such as performance of science missions, may require a much

less aggressive approach. Discussions with NASA-Goddard engineers suggest that Leonardo

formation-keeping operations might occur every two weeks.? For this reason, we simulate

27

a maneuver from initial conditions defined by two weeks of secular J2 drift. These initial

conditions are calculated by applying drift to the appropriate spacecraft orbital elements,

and recalculating the formation master orbit and desired orbital states from the perturbed

formation states.

By calculating the master orbit for this maneuver from the perturbed orbit states, we

can re-center the formation, so that the bulk drift is ignored, and only the differential secular

drift is corrected.

Figure 5 illustrates the simulation of a two-week-J2-error formation-keeping maneuver,

using classical orbital element Lyapunov control scaled to 0.01 N. The figure shows the

orbital-frame components of position error for all six spacecraft in the formation. As before,

we use gain scheduling, with an emphasis on the out-of-plane error in the first stage of the

maneuver, and the in-plane angular error in the second stage. We define the end of the first

stage, and the beginning of the second stage as the point where the control law no longer

requests thrust for 30 consecutive time steps. At this time, the elemental errors are within

the error envelope, and mean motion control commences. Notice that gain scheduling is

applied on different schedules for each of the spacecraft. In fact, individual gains should be

calculated for each spacecraft using the methods described in this paper.

Table 3 shows the initial values of δΩj and δθj, and the ∆V required to re-establish

the formation after two weeks of J2 drift. The results show a significant imbalance in fuel

consumption, with spacecraft 3 and 4 performing the majority of the thrusting. This result

is not surprising, as these spacecraft must overcome the largest initial elemental errors.

Comparing spacecraft 1 with spacecraft 2, we notice that while spacecraft 1 starts with

larger initial elemental errors, it requires fewer thrust impulses. This anomaly is actually

the result of the original equations of motion. The time rate of change of Ω as a result of

orbit normal thrust is

dΩ

dt=

r sin θ

h sin iuθ (61)

Notice that decreasing inclination results in increasing Ω. Therefore, spacecraft with lower

28

inclinations require less thrust to correct the same Ω errors. This observation suggests that

a shift of the master Ω towards the highly inclined orbits may reduce the total thrust use.

Table 4 shows the results of a formation-keeping maneuver with δΩ values shifted by 0.1

toward the spacecraft with higher inclinations. The results of this shift are significant: while

there is little improvement in the thrust distribution (cf. Table 3), the total thrust count

decreases by almost 10%. Furthermore, the thrust count decreases for every spacecraft in

the formation, including those whose initial orbital element errors increase as a result of the

Ω shift. The cause of this result is simple: by shifting the right ascension angle of the master

point (and thus the target points, which are defined from the master orbit), the target orbit

is moved closer to the specific spacecraft.

This effect is illustrated in Fig. 6. Notice that an increase of the right ascension error

magnitude results in a decrease in the position error magnitude. The target point (‘’) in the

orbit rotated the most about the inertial z-axis (up), is much closer to the spacecraft (‘¦’)

than is the target point in the orbit rotated less about the z-axis. Because the element θ is

measured from the node (‘•’), and because the orbit is inclined so little, increasing the angle

between the target node and the actual node, while holding δθ constant, actually brings

the target point closer to the spacecraft. As a result, less thrust is required to perform the

maneuver.

This effect is also illustrated by the results of previous simulations. If we had plotted

the in-plane angle error during the formation-establishment maneuver, we would see that

during the first phase of the maneuver, where the selected gains correct the out-of-plane

error, the value of δθ decreases dramatically. This correction of the in-plane angular error

is not a result of mean motion control, but a result of the movement of the node, the point

from which θ is measured. As the δΩ error is corrected, the δθ error decreases, because the

points from which θ and θ∗ are measured get closer together. This observation is a strong

argument in favor of using gain scheduling for this control. For this formation (and others

with low inclinations), it is unwise to allow feedback of δθ initially, as the points from which

29

θ and θ∗ are measured can vary greatly until the spacecraft has acquired the appropriate,

target orbital plane.

Conclusions

We develop nonlinear orbit control laws for use in spacecraft orbital maneuvers, and space-

craft formation flying. This nonlinear orbital element feedback control law, augmented by

a new mean motion control strategy, successfully and efficiently maintains and modifies the

full Keplerian motion of a large variety of spacecraft orbits. Unlike some previous element

feedback controls, this law controls both the orbit size, shape, and orientation, and the space-

craft’s angular position within the orbit. The control also takes advantage of physical insight

to minimize fuel consumption, by utilizing differential mean motion to correct in-plane angle

errors. A new technique for choosing orbital element feedback gains appropriately scales

the gains to provide efficient, chatter-free thrust performance. The master orbit algorithm

serves as a starting point for a formation-establishment maneuver, and also a point from

which we can calculate the desired spacecraft orbital elements. Simulation results show that

for the Leonardo formation, the definition of the master orbit at the geometric center of

the formation does not provide fuel optimal results. Future studies should seek other ana-

lytical methods for calculating a fuel optimal formation master orbit. Another interesting

conclusion drawn from the simulation results is that the total thrust required to complete

a maneuver is not necessarily proportional to the initial elemental errors. Because of the

interaction between the elements, increasing some elemental errors might actually result in

decreased thrust use. This observation suggests that the geometric center of the formation

is not the best choice for a master reference orbit, and that the master formation orbit

definition should be refined.

Acknowledgements

This work was supported by Arje Nachman of the Air Force Office of Scientific Research

and by Jesse Leitner of NASA Goddard Space Flight Center. The authors are grateful to

30

Matthew Berry, Christopher Karlgaard, and Craig Woolsey of the Department of Aerospace

and Ocean Engineering at Virginia Tech for many useful discussions regarding this problem.

31

References

[1] Ezell, E. C. and Ezell, L. N., “The Partnership: A History of the Apollo-Soyuz Test

Project,” The NASA History Series , 1978, NASA SP-4209.

[2] Folta, D. and Hawkins, A., “Preliminary Results of NASA’s First Autonomous For-

mation Flying Experiment: Earth Observing-1 (EO-1),” Flight Mechanics Symposium,

Goddard Space Flight Center, Greenbelt, MD, 2001.

[3] Schaub, H. and Alfriend, K. T., “J2 Invariant Relative Orbits for Spacecraft Forma-

tions,” Celestial Mechanics & Dynamical Astronomy , Vol. 79, No. 2, 2001, pp. 77–95.

[4] Wiesel, W. E., “Relative Satellite Motion About an Oblate Planet,” Journal of Guid-

ance, Control, and Dynamics , Vol. 25, No. 4, 2002, pp. 776–785.

[5] Wielicki, B. A., Cess, R. D., King, M. D., Randall, D. A., and Harrison, E. F., “Mission

to Planet Earth: Role of clouds and radiation in climate,” Bulletin of the American

Meteorological Society , Vol. 76, 1995, pp. 2125–2153.

[6] Hughes, S. P. and Mailhe, L. M., “A Preliminary Formation Flying Orbit Dynamics

Analysis for Leonardo-BRDF,” Flight Mechanics Symposium, NASA Goddard Space

Flight Center, Greenbelt, MD, 2001.

[7] Vassar, R. H. and Sherwood, R. B., “Formationkeeping for a Pair of Satellites in a

Circular Orbit,” Journal of Guidance, Control, and Dynamics , Vol. 8, No. 2, 1985,

pp. 235–242.

[8] Leonard, C. L., Hollister, W. M., and Bergman, E. V., “Orbital Formationkeeping with

Differential Drag,” Journal of Guidance, Control and Dynamics , Vol. 12, No. 1, 1989,

pp. 108–113.

[9] Middour, J. W., “Along Track Formationkeeping for Satellites With Low Eccentricity,”

The Journal of the Astronautical Sciences , Vol. 41, No. 1, 1993, pp. 19–33.

32

[10] Folta, D. C. and Quinn, D. A., “A universal 3-D method for controlling the relative

motion of multiple spacecraft in any orbit,” AIAA/AAS Astrodynamics Specialist Con-

ference, Boston, MA, 1998, AIAA 98-4193 (A98-37403).

[11] Carpenter, J. R., Folta, D. C., and Quinn, D. A., “Integration of Decentralized Linear-

Quadratic-Gaussian Control into GSFC’s Universal 3-D Autonomous Formation Flying

Algorithm,” AIAA Meeting Papers on Disc, Vol. 4, AIAA, Washington, DC, 1999.

[12] Carpenter, J. R., “Feasibility of Decentralized Linear-Quadratic-Gaussian Control of

Autonomous Distributed Spacecraft,” Flight Mechanics Symposium, Goddard Space

Flight Center, Greenbelt, MD, 1999.

[13] Schaub, H., Vadali, S. R., Junkins, J. L., and Alfriend, K. T., “Spacecraft Formation

Flying Control Using Mean Orbital Elements,” Advances in the Astronautical Sciences ,

Vol. 103, Part 1, 1999, pp. 163–181, AAS 99-310.

[14] Schaub, H. and Alfriend, K. T., “Impulsive Spacecraft Formation Flying Control to

Establish Specific Mean Orbit Elements,” Journal of Guidance, Control, and Dynamics ,

Vol. 24, No. 4, 2001, pp. 739–745.

[15] Ilgen, M. R., “Low Thrust OTV Guidance Using Lyapunov Optimal Feedback Control

Techniques,” Advances in the Astronautical Sciences , Vol. 85, Part 2, 1993, pp. 1527–

1545, AAS 93-680.

[16] Chang, D. E., Chichka, D. F., and Marsden, J. E., “Lyapunov-Based Transfer Be-

tween Elliptic Keplerian Orbits,” Discrete and Continuous Dynamical Systems–Series

B , Vol. 2, No. 1, 2002, pp. 57–67.

[17] Naasz, B. J., Classical Element Feedback Control for Spacecraft Orbital Maneuvers ,

Master’s thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA,

May 2002.

33

[18] Bate, R. R., Mueller, D. D., and White, J. E., Fundamentals of Astrodynamics , Dover

Publications, Inc., New York, NY, 1971.

[19] Battin, R. H., An Introduction to the Mathematics and Methods of Astrodynamics,

Revised Edition, American Institute of Aeronautics and Astronautics, Inc, Reston, VA,

1999.

[20] Sedwick, R., Miller, D., and Kong, E., “Mitigation of Differential Perturbations in Clus-

ters of Formation Flying Satellites,” Advances in the Astronautical Sciences , Vol. 102,

Part 1, 1999, pp. 323–342, AAS 99-124.

[21] Vadali, S. and Vaddi, S., “Orbit Establishment for Formation Flying Satellites,” Ad-

vances in the Astronautical Sciences , Vol. 105, Part I, 2000, pp. 181–194, AAS 00-111.

[22] Hughes, S. P. and Hall, C. D., “Optimal Configurations for Rotating Spacecraft Forma-

tions,” Journal of the Astronautical Sciences , Vol. 48, No. 2–3, 2000, pp. 225–247.

[23] Schaub, H. and Alfriend, T., “J2 Invariant Relative Orbits for Spacecraft Formations,”

Flight Mechanics Symposium, NASA Goddard Space Flight Center, Greenbelt, MD,

1999, pp. 125–139.

34

Table Captions

Table 1: Leonardo formation elemental offsets

Table 2: Leonardo formation orbital elements

Table 3: Leonardo formation-keeping

Table 4: Leonardo formation-keeping with δΩ shifted by 0.1

35

Table 1: Leonardo formation elemental offsets(a = 6778 km, e=0)

s/c i[] ∆Ω[] ∆θ[]1 0.4783 16.86 -9.992 0.9771 -20.88 25.523 4.4108 166.10 -160.404 4.5866 17.42 -13.585 0.6153 17.52 -14.666 2.4481 -157.80 160.13

Table 2: Leonardo formation orbital elements(a = 6778 km, e=0)

s/c i[] Ω[] θ[]1 0.4783 16.86 -9.992 0.9771 -20.88 25.523 4.4108 166.10 -160.404 4.5866 17.42 -13.585 0.6153 17.52 -14.666 2.4481 -157.80 160.13

master 0.2497 98.24 266.14

36

Table 3: Leonardo formation-keeping

s/c i[] δΩ[] δθ[] ∆V [m/s]1 0.4783 -0.1335 1.0667 3.16272 0.9771 -0.1211 0.9670 3.19403 4.4108 0.1965 -1.5693 6.83204 4.5866 0.2236 -1.7857 7.80605 0.6153 -0.1310 1.0461 3.38006 2.4481 -0.0346 0.2753 1.6187

TOTAL 25.9934

Table 4: Leonardo formation-keeping with δΩ shifted by 0.1

s/c i[] δΩ[] δθ[] ∆V [m/s]1 0.4783 -0.2335 1.0667 3.15202 0.9771 -0.2211 0.9670 3.31533 4.4108 0.0965 -1.5693 5.51404 4.5866 0.1236 -1.7857 6.52735 0.6153 -0.2310 1.0461 3.31276 2.4481 -0.1346 0.2753 1.9413

TOTAL 23.7626

37

Figure Captions

Fig. 1: Formation simulation models: A) target-interceptor model; B) leader-follower model;

C) cooperative model

Fig. 2: Combination orbit-raising and inclination shift maneuver. A) Ka = 1, Ke = 1, and

Ki = 50 B) Ka = 1, Ke = 1, and Ki = 5

Fig. 3: Formation establishment using 1-N scaled elemental feedback control with gain

scheduling

Fig. 4: Evolution of semi-major axis during mean motion controlled formation-establishment

maneuver

Fig. 5: Two week J2 perturbed formation-keeping using elemental control with gain schedul-

ing

Fig. 6: Increasing right ascension error maps to decreasing position error

38

δr δr δr

δr

δν δν

A B C

– passive spacecraft

– reference orbit

– active spacecraft

Figure 1: Formation simulation models: A) target-interceptor model; B) leader-followermodel; C) cooperative model

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−50000

5000

−50000

5000

−5000

0

5000

r1 [km]r

2 [km]

r 3 [km

]

−50000

5000

−50000

5000

−5000

0

5000

r1 [km]r

2 [km]

r 3 [km

]Initial OrbitControlled OrbitTarget Orbit

A B

Figure 2: Combination orbit-raising and inclination shift maneuver. A) Ka = 1, Ke = 1,and Ki = 50 B) Ka = 1, Ke = 1, and Ki = 5

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0 2 4 6 8 10−100

−50

0

50

100

150

200

250

300

Orbits

Pos

ition

err

or [k

m]

r1

r2

r3

Figure 3: Formation establishment using 1-N scaled elemental feedback control with gainscheduling

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0 2 4 6 8 106775

6780

6785

6790

6795

6800

Orbits

Sem

i−m

ajor

axi

s [k

m]

aa*

a**

Figure 4: Evolution of semi-major axis during mean motion controlled formation-establishment maneuver

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0 10 20−200

0

200a) Satellite 1

[km

]

0 10 20−200

0

200b) Satellite 2

[km

]0 10 20

−200

0

200c) Satellite 3

[km

]

0 10 20−200

0

200d) Satellite 4

[km

]

0 10 20−200

0

200e) Satellite 5

[km

]

0 10 20−200

0

200f) Satellite 6

[km

] δr1

δr2

δr3

Figure 5: Two week J2 perturbed formation-keeping using elemental control with gainscheduling

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−5000

0

5000−6000 −4000 −2000 0 2000 4000 6000

0

1000

2000

3000

Node Crossingδν = 0Target PointSpacecraft

Figure 6: Increasing right ascension error maps to decreasing position error

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